A dielectric theory of “multi-stratified shell” model with its application to a lymphoma cell

J. theor. Biol. (1979) 78, 251-269

A Dielectric Theory of “Multi-Stratified Shell” Model with its Application to a Lymphoma Cell IRIMAJIRIt Department of Physiology, Kyoto University School of Medicine, Sakyo-ku, Kyoto 606, Japan TETSUYAHANAI Institute for Chemical Research, Kyoto University Uji, Kyoto-fu 611, Japan AKIHIKO

AND AKIRA INOUYE

Department of Physiology, Kyoto University School of Medicine, Sakyo-ku, Kyoto 606, Japan (Received 8 June 1978, and in revisedform

15 November 1978)

A theory of complex dielectricconstant (a*) for the suspension of “multistratified” sphericalparticlesis presented.Basedon Maxwell’s theory of interfacial polarization, we derive a generalexpressionwhich correlatesa* with the electrical and geometricalparametersof eachstratum. It can be shown that such a “multi-stratified” systemin generalshould give rise to multiple dielectric dispersions,the number of which correspondsto the number of interfaceslying betweenthe successive shell phases.The conditions for a full number of different “unit” dispersionsto occur are also discussed.As an example, a special case of the “double-shell” model consistingof a sphericalcore and threelayersof concentricphasesis solved numericallyby usinga setof parametervaluespertinent to a lymphomacell. In light of the characteristicbehavior of a* thus revealed,we proposea schemeof procedurethat appliesto the determinationof electrical parametersassociatedwith the specific“double-shell” model. 1. Introduction As described in our previous report (Irimajiri, Doida, Hanai & Inouye, 1978, hereafter, Paper I) the suspension of cultured lymphoma cells displayed somewhat complicated dielectric behavior when measured in a frequency range of 0.01-100 MHz. In an attempt to interpret such au observation we coined a model named the “double-shell” model and employed it instead of the conventional “single-shell” model (Pauly & Schwan, 1959; Hanai, Koizumi & Irimajiri, 1975). Introduction of this new model was considered advantageous in that, within the outer shell (i.e. the plasma membrane), i Present address: Department of Physiology, Kochi Medical School, Nankoku, Kochl 781-51, Japan. 251 0022-5193/79/100251+ 19 $02.00/0 0 1979 Academic Press Inc. (London) Ltd.

A.

252

IRIMAJIRI

ET

AL.

another shell phase (i.e. the nuclear membrane) demarcates a mass of nucleoplasm from the rest of cytoplasm in the lymphoid cell interior. In fact, such an expectation was fulfilled as far as the results of our curve-fitting method were concerned (cf. Paper I, Figs 5 and 7). In Paper I, however, we employed rather intuitively a mathematical expression to calculate complex dielectric constants for the “double-shell” model. Its theoretical basis will be substantiated in the present paper. Fricke (1955) developed a similar theory for the complex conductivity of stratified particles in suspension; he derived an admittance equation of the form of a continued fraction based on a ladder-type network of the complex conductivities that represent the successive strata. Following Fricke’s approach did not seem to be much more helpful for the purpose of determining the model’s phase parameters, however. In the present communication, therefore, we shall (i) write down a general expression for the dielectric constant of the “multi-stratified” system, (ii) point out some characteristics of its dispersive behavior, and (iii) put forward a scheme of procedure through which one may assign, given raw data of dielectric measurements, a plausible set of parameter values pertinent specifically to the “double-shell” model. 2. Theory (A)

LIST

OF SYMBOLS

E* = complex dielectric constant of suspension, E* = E+ K/$X, E = dielectric constant IC= conductivity (S m- ‘) E, = permittivity of vacuum (= 8.85 x 10-l’ F m-‘) j=JT 0 = angular frequency (= 2$) EX = complex dielectric constant of suspending medium 5: = equivalent, homogeneous, complex dielectric constant of ith sphere ET = complex dielectric constant of ith shell (i = 1,2, . . . , n) or of spherical core (i = Iz + I), ET = El+ ICiljOE~ CD= volume concentration of suspension Ri = radius of ith sphere Pi(z), Qi(z), pi(z), ci(z) = polynomials = IT i&i*- 1

Xi

1; = ET/Ef- 1 Aj

=

z=jo

&j/&j-

1

of z of degree i

pi = Ici/lij- 1

aj = 2(lh4i)/(l+2+i)

pi = lci/Eiev

bi

=

C2+4i)lC1

4i

ci

=

(l

=

(Ri+

l/&l3

+24i)/t1

-(bi> -4i>

“MULTI-STRATIFIED (B)

GENERAL

EXPRESSION

SHELL” FOR

COMPLEX

MODEL

DIELECTRIC

253 CONSTANT

&*

Let us suppose a suspension of spherical particles (dielectric constant, FT; radius, R,) uniformly dispersed in a medium of so*to a volume concentration Q which is small enough to neglect the mutual polarization effect between the suspended particles [Fig. l(a)]. According to the theory of heterogeneous systems (Maxwell, 1891; Wagner, 1914; for review, see e.g. Hanai, 1968; Dukhin, 1971), the equivalent, homogeneous dielectric constant E* for the whole suspension is given by E* = F* 2&o*+ ET - 2@(&5- $j "

2E;j++7+@(E;-iT)

,2(1-@)+(1+2@)$/E; (1) = Eo (2+@)+(1 -@)$/ET, If we assume that equation (1) holds for any suspension system in which the dispersed particles can be virtually regarded as homogeneous and having a dielectric constant ET but actually have a stratzjied structure such as shown in Fig. l(b), then the problem will be reduced to expressing the temporarily assigned parameter ET in terms of ET'S and Ri’s with i = 1,2, . . . , IZ+ 1. To begin with, we shall again assume that the entire sphere of radius RI can be subdivided into two phases: the outermost shell of thickness R, -R, having an actual dielectric constant ET and the spherical core of radius R, having an equivalent, homogeneous dielectric constant Fz which, for the

(a ) (b) 1. A “multi-stratified shell” model. (a) Homogeneous spheres (radius R, , complex dielectric constant E;*) suspended in a medium (complex dielectric constant agt) to a volume concentration Q, give rise to an equivalent, homogeneous, complex dielectric constant E*. (b) Each particle in (a) consists of n concentric shells (outer radii Ri's, complex dielectric constants Q’s) and the core phase (radius R,, 1,complex dielectric constant E,*+ i). ET stands for the equivalent, homogeneous dielectric constant of the smeared-out, spherical domain of radius Ri. FIG.

254

A.

IRIMAJIRI

ET

AL.

moment, is left to be embodied by the further steps of subdivision. It follows then, as was suggested by Maxwell (1891) that one may uniquely assign Ez so as to satisfy the relation: F* = E* W-41)+(1 +v4w~T ‘1

-~lE/~:

(2+4d+U

l

where +i is the fractional volume of the second sphere relative to the first, or 4i = (R2/R,)3. Quite similarly, the equivalent, homogeneous dielectric constant for any ith sphere inside the whole sphere is given by p? ‘1

=

FI* 2(1-4i)+(1 ‘1

+24i)ET~l/F.:,

(i

=

1

2

. .

n)

(2) ’ ’ .’ (2+44+(l-C$&~&J$ where EF+1 refers to the successively smeared-out subphases and $i = (Ri+,/Ri)“. Since F,*+, = E,*+~ for the innermost core: we can derive an expression for ET by tracing back the above relations from i = n to i = 1. The combination of equations (1) and (2) then enables us to reach the sought-for, general formula of E* through which one may predict the frequency dependence of E and K starting with the knowledge of Ei’S, Ki’S, Ri’S and @. (C)

ANALYSIS

OF THE

GENERAL

FORMULA

The ultimate goal of this section is to derive an explicit equation for E* as a function ofjw, thereby making it easier to reach a physical implication of the formula obtained above. To this end it appears convenient to rewrite equation (2) first. Dividing equation (2) by ET-1, we have Et Ef- 1

ET 2(lL~~)+(l+2~i)F~+“+,/E~ ET-1 (2+$i)+(l-$i)EF+i/&i*

which may be simply rewritten in the recurrence form xi

=

Here we have introduced xi =

c,/z,*

ai+xi+l _

. I t?,+x,+,

3 (i = 1,2. . . . ) n). \

’

(3)

the following definitions:

E,*+~/E,* (for i = n+l) 1 q/&i*- 1

(4)

(5) (6)

Ui = 2( 1- #i)/( I+ 24!Ji) (for i = 1, 2, bi

=

C2+4i)/t1

Ci = (1+24i)/(l

-4i) - Cji).

(7)

“MULTI-STRATIFIED

SHELL”

255

MODEL

Note that the constants, ai, b, and Ci, in (7) should satisfy bi>2>Ui>O

and

Ci>l

(8)

because, by definition, 0 CC& < 1. Next, from the definition equation (6) may be transformed into

of ET and E?- 1,

A: = &(Z+pi)/(Z+pi-1)

(9)

where ii = EilEi- 1, pi = KileiE,, and z = Jo. Hence the condition remain a complex fraction is given by or

Pi

f

Pi-1

(i=1,2

$/&i # ui-l,iEi-l

,...)

n).

for A: to

(10)

Otherwise, ,X: reduces to a real number li. The same is also true with X, + 1; the condition for its being complex is given by inequality (10) for i = n + 1. In the analysis to be presented below, we shall consider mainly the systems in which condition (10) holds for all i’s including i = n + 1. Now let us begin by examining the nth phase (i.e. the innermost shell) in contact with the neighboring ones (i.e. the core phase and the n- lth shell). Equation (3) for i = n is written, by taking into account relation (4), as

By introducing parameters, u and p, defined in Appendix A to be independent of z ( =jw), and substituting equation (9) for i = n into equation (1 l), x, as a function of z may be expressed in the form:

(z+p,)(z+4 xn=ch(z+pn-1)(z+P) where CL is a constant. It can be shown (see Appendix A) that the three parameters in the equation above are bound by the constraint: a’tBPP”

(13)

P.-l

(14)

so that, unless = 4

equation (12) may be further rewritten as

,(z+Pdz+Pz) __P2(-4 -?I=cn(z+q1)(z+q2) =cLez(z)

(15)

256

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ET

AL.

where

p1= min(P,,4,

~2

=

max

q1

q2

=

maxh-,,

=

min

(P,-

1, P).

In passing, it should be noticed that, if condition instead of expression (15)

(P,,

4 8).

(14) holds, we merely have,

x, = c’,(z+pn)l(z+P~ = 4 plwQlw even when the condition

(16)

(17)

(10) is satisfied, viz., Pn-1 f P” + Pn+1.

(18)

As is inferred from relation (15) and will be proved in Appendix B, the final form of xi referring to the whole sphere is generally given by , (z+Pdz+Pz) x1

=

ci

*. . (Z+Pn+i)

(z+qJ(z+q2).

~

c;

p”+lw

Q,+I

. .(z+q,+1)

(19)

where P,,+ 1(z) and Q,,+ i(z) are polynomials prime to each other with pl , ql, pz, and q2 differing in value from those of (15), and c; is a constant. On the other hand, equation (1) may be rewritten as e* = CEO*(a+Xl)/(b+Xl)

(20)

where a=2-,

l--CD 1+2
b-2+4’ l-at

and

c=E,

and hence, for 0 < @< 1, we have b>a>O

Substituting

and

c>l.

(21)

equation (19) together with .sg = .co(z+po)/z, into (20) gives F* = c,z+~~ __. z

Pn+~(z)+a’Qn+l(4 P,+,(z)+b’Qn+,(z)

(22)

where a’, b’ and c’ are again constants independent of z and b’ > a’ > 0. Now, with the aid of Appendix C, equation (22) can be shown to take the form: (23)

“MULTI-STRATIFIED

SHELL”

MODEL

257

Here pk’s and qk’s are not the same as those of (19) and satisfy the sequence : 0 =

41
...

(24)

<9n+2
In principle, equation (23) is further transformed, by means of partial-fraction decomposition, into the form:

n+ 2

= const+ C ~ k=Z

rkhk

1 fzhk

I rlh ZF”

(25)

where the constants rk’s satisfy (see Appendix D) r,>O

(k = 1,2,. ..,n+2).

(26)

Changing the notation hitherto used, we finally obtain an expression for F* : (27)

Inspection of the final equation thus obtained reveals that the new symbols introduced above must have the following physical meanings: E = limiting K = limiting

dielectric constant at high frequencies (Ed) conductivity at low frequencies (K~) rk = relaxation time of kth “Unit” dispersion L, = dielectric increment associated with the kth “unit” dispersion (Ask). The results of the foregoing analysis may be summarized as follows: (a) A dilute suspension of “multi-stratified” spherical particles in general gives rise to a composite dielectric dispersion which can be decomposed into its “unit” dispersions characterized by the respective relaxation times zk’s. (b) The maximum number of such “unit” dispersions corresponds to the number of interfaces each of which demarcates the dielectrically distinguishable subphases within the suspended particles. In other words, given a system of n shells that successively enwrap the central core, we can expect, theoretically at least, n + 1 different relaxation times to be mvolved in the general case. (c) The principal condition for a full number of the possible relaxations to occur is that pi#pi-1 (i= 1,2,..., n + 1) ; this condition alone is by no means sufficient, however. As will be detailed in Appendix B [see specifically equations (B13) and (BlS)], another condition is necessary and is given by pi-r#pk(i=l,2 ,..., n;k=l,2 ,..., n+l-i)Where~kisaverycomplicated function of the phase as well as the size parameters and is defined by

258

A.

IRIMAJIRI

ET

AL.

equations (B14) and (B15) in Appendix B. A corollary will be : Degeneracy of “unit” relaxations may well take place whenever either of the above conditions fails to be met. 3. Comments on Individual

Cases of the Generalized Model

Case n = 1 (“single-shell” model) The dielectric behavior of the whole suspension is described by the combination of equations (1) and (2) for n = 1. Pauly & Schwan (1959) worked out this case to present a rigorous solution and derived some approximate formulae convenient for a practical data analysis. An elaborate discussion as to the applicability of this model has been added also by Hanai et al. (1975) on the basis of the Pauly-Schwan expression. However one point, though trivial, appears to have escaped the previous authors’ notice; that is, as stated under item (c) of the preceding section, if relation (14) holds, viz., KO -zz EO

alrcl + ti2 alq +E*

then the “single-shell” model can only give rise to oozerelaxation time even when rco/eo # rcl/sl # ~,/e~, i.e. condition (10). At any rate, the “single-shell” model is claimed to apply to a wide variety of spherical “shell” particles including cells, subcellular organelles like isolated mitochondria (Pauly, Packer & Schwan, 1960) and synaptosomes (Irimajiri, Hanai & Inouye, 1975b), and unilamellar lipid vesicles (Redwood, Takashima, Schwan & Thompson, 1972). Case n = 2 This type of model may simulate a certain species of micro-organisms having an additional “shell” phase (e.g. the cell wall) outside their plasma membrane. Dielectric properties of yeast cells (Asami, Hanai & Koizumi, 1976) and E. coli (Carstensen, Cox, Mercer & Natale, 1965 ; Carstensen, 1967) have been studied in light of this model with y1= 2. To raise another point, one must have recourse to this model when one deals with a surface-charge layer as a separate phase extending to a definite thickness from the boundary of an ordinary lipid-membrane matrix. Case n = 3 (“double-shell” model) By this type would be best represented the category of lymphoid cells whose nuclei are round and large enough to occupy more than 4 of the intracellular space. To better illustrate an immediate need for the “double-shell” model in the analysis of cell interior, it appears helpful to visualize the size effect due to

“MULTI-STRATIFIED

SHELL”

MODEL

10 -

259

- 9.5

“:

Z‘E

0 ;

:

w

Y 5-

o0.01

-

I 0. I

/ I Frequency

I IO

9.0

-8.5 100

(MHZ)

2. Effect of the size of the inner shell upon the overall dielectric constant (8) and conductivity (K), as functions of frequency (f), for the “double-shell” model. Calculation was made with the following parameter values: (Note that subscripts should read: a = 0, s = 1, c = 2, n=3, and k=4.) ~,=~,=77, &,=9, &,=20, E~=~OO, ~,=1~,=~~=10rnScrn-~, K, = 8 x lo-’ mS cm-‘, K, = 4 x 1O-3 mS cm-‘, RI = 65pm, RI-RZ=80P\, R,-R,=400& and Rx = 0 (-), 23 (---), 4.67 (------), or 6.25 (. .) pm; and @ = 0.1. FIG.

the internal “shell” body. The behavior of E* in question is predictable from numerical calculation using equation (1) combined with equations (2) for i = 1,2, and 3. Figure 2 shows the calculated dispersion curves for a hypothetical cell system in which the nuclear volume was varied from nil up to 90 “/I of the total cell volume. Clearly, deviation from the initial pattern (solid curues) referring to the “no nucleus” (or the “single-shell”) case developed with increase in the nuclear volume. Such a finding points to a tentative generalization that if the parameter values are as specified in the legend to Fig. 2 then the “single-shell” model will be a poor approximation for nucleate cells whose nuclear volume exceeds 6% of the intracellular space. With our specimen, lymphoma L5 178Y (cf. Paper I), the mean nuclear volume fell near 40 %, so that the introduction of a new model other than the conventional one was pertinent or rather necessary for a successful interpretation of the experimental dispersion curves. 4. The “Double-Shell” Model (A)

CHARACTERISTIC

BEHAVIOR

AS REVEALED

BY APPLICATION

TO A LYMPHOMA

CELL

It seems helpful to examine the model’s dielectric behavior, as a function of individual phase parameters, prior to getting into the problem of how to

A.

260

IRIMAJIRI

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AL.

evaluate those parameters starting from the observed dispersion curves. In Fig. 3 are shown 8 sets of dispersion diagrams thus obtained. Here, the set of the phase and size parameters comprising the reference state has been chosen from the specimen we used in Paper I, and computation was carried out by varying only one “probe” parameter (denoted by the encircled symbols) with the other parameters fixed at values for the reference state. In this diagram, though the scope of coverage was not so comprehensive but limited to a lymphoma cell, still we can extract some general features that are associated with the “double-shell” model. The characteristics found are summarized into the following items (i)-(iv).

/0 / IO.0 ,a--------20 ,._ _._ . ~5 _ LO ._._..... @cm-’ ,. / ; ‘;;> m 100 o-01 0.1 I IO loo 9.0

ol-/ 0.01

L, 0-I

I

IO

)

J

‘c&&J

Frequency(MHz)

3. Calculated dielectric behavior of the “double-shell” model. In each figure, only theprobe parameter (denoted by the encircled symbols) was changed with all the others fixed at the reference state defined as: E, = E, = Ed = 77, E, = 9, E, = 40, K, = K, = K~ = IO mS cm-‘, K, = R3 = 467 pm, 8 x lo-’ mS cm-‘, K, ‘- 4 x 10e3 mS cm-‘, R, = 6.5 pm, R, - R, = 80 A, R3 - R, = 400 ,k, and @ = 0.1. Curves for the reference state are indicated by solid lines. Symbols are as in Fig. 2. FIG.

“MULTI-STRATIFIED

SHELL”

261

MODEL

For simplicity, let us use the symbols: E[, E,,,I+, rc,I-subscripts I and h indicate the limiting values at low (f< 0.1 MHz) and high (f> 10 MHz) frequencies. fijz : “apparent” ‘relaxation frequency defined as (8 at fi,J - sh = (Ed- sJ2

f ( 1,2,f, ,,2 : frequency region below or above fiiz.

Subscripts, s, c, n and k, to E or K in the “double-shell” model refer respectively to plasma membrane (or outer shell), cytoplasm, nuclear envelope (or inner shell) and karyoplasm. (i) The low-frequency behavior, sr and rc!, are primarily determined by the outer-shell parameters, E, and K,. (ii) The high-frequency dielectric constant shis affected by E, or sk, while the high-frequency conductivity rch depends strongly on rc, and Kk, weakly on E, and Ed, and almost negligibly on the others. (iii) Given fixed values of E, and K,, the major factor that may cause a shift in fi,2 is ICY,less influencing are rcn. E, and r$. (iv) Given E,, rc, and rcC, then the slope of dispersion curves around fi!2 changes depending upon IC,,, E, and r$. As regards the predictable number of the “unit” relaxations (or relaxation times, zk’s) one could locate at least three out offour relaxations that must be

involved in the “double-shell” model, in the case of K, = 2 mS cm-’ (Fig. 3, @ ) as well as in the case of Rs = 6.25 urn (Fig. 2) ; the fourth relaxation might be traced up in the frequency region far above 100 MHz. For the other combination of the phase parameters, however, such a separation in the relaxation frequencies appears to be more or less obscured as far as the results of Figs 2 and 3 are concerned. (B)

A PROCEDURE

FOR

DETERMINING

THE

PHASE

PARAMETERS

On the analogy of the method advanced for the “single-shell” er al., 1975), a stepwise procedure using a curve-fitting

model (Hanai

method appears most

effective also in the “double-shell” case. When a pair of dispersion and K vs. frequency are available together with the size parameters and the external parameters (E, and tiJ, one may go through the steps, guided by the aforementioned items (i)-(iv), to find the most set of the sought-for “phase parameters”. Step 1. To start computation, enter an interim set of parameters &, = E, = Ek, (here, let E, satisfy s,s,/(Rr -R,)

7c,= rc, = lck3 E, = &n = 1 uF cm-“) and

K, = K, = 1W6 mS cm-’

data on E (i.e., R,‘s) following plausible such that

262

A.

IRIMAJIRI

ET

AL.

(this value corresponds to a membrane resistance of 1 kQ cm2 for a membrane thickness of 100 A). For entering QD,a reliable value can be estimated through the simple relation (Irimajiri, Hanai & Inouye, 197%): CD= 1 -(KJK,)1’5

(28)

provided that rc,s- K, x 1o-s. Step 2. Confirm whether the pre-set value of 0 well explains the observed G* Step 3. Search and fix E, so that aI calculated may fit the observed Q. Step 4. Search and fix K, so that fli2 may fit the observed value of fiiZ. Step 5. Search and fix K, so that the shoulders (atf< 1,2 for Eand atf, 1,2 for

K) may duplicate those observed. Step 6. Search and fix JC~so that both rch (or rc at f, 1,2) and the settling portion of an e-curve at f, 1,2 may trace those observed. Step 7. Search and fix E, and Q, likewise, to attain a better fit in the frequency regions around and above fi,2. Usually, more than 4 iterations of the above Steps were required for the best fit. A further improvement would be feasible by means of a computerized “non-linear” least squares fitting method. Without recourse to it, however, a fairly satisfactory result has been obtained, as demonstrated in Paper I, by following the procedure presented above. Another comment on this procedure is : As far as the specific lymphoma cell here examined is concerned, it was rather difficult to fix ak separate from Ed, and vice versa, because the contributions of both parameters to the final shape of dispersion curves resembled each other at least in the frequency range up to 100 MHz. A remedy to such a problem would be either to assume a reasonable value for one of the two, as was made in Paper I, or to extend the measurements far over 100 MHz where the microwave technique (cf. Schwan, 1963) will be required. We wish to thank Dr N. Koizumi and Dr Y. Doida for helpful suggestion. A. Irimajiri acknowledges support from the Buswell Fellowship, State University of New York at Buffalo, in tenure of which part of the present study was performed. REFERENCES ASAMI, K., HANAI, T. & KOIZUMI, N. (1976). J. mem. Biol. 28, 169. CARSTENSEN, E. L., Cox, H. A., Jr., MERCER, W. B. & NATALE, L. A. (1965). CARSTENSEN, E. L. (1967). Biophys. J. 7, 493. DUKHIN, S. S. (1971). Surface Colloid Sci. 3, 83.

Biophys.

J. 5,289.

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FRICKE, H. (1955). J. phys. Chem. 59, 168. HANAI, T. (1968). In Emulsion Science (P. Sherman, ed.), ch. 5. London: Academic Press Inc. HANAI, T., KOIZUMI, N. & IRIMAJIRI, A. (1975). Biophys. Struct. Mechanism 1, 285. IRIMAJIRI, A., DOIDA, Y., HANAI, T. & INOUYE, A. (1978). J. mem. Biol. 38, 209. IRIMAJIRI, A., HANAI, T. & INOUYE, A. (1975a). Experientiu. 31, 1373. IRIMAJIRI, A., HANAI, T. & INOUYE, A. (1975b). Biophys. Struct. Mechanism. 1, 273. MAXWELL, J. C. (1891). A Treatise on Electricity and Magnetism, 3rd edn., Vol. I, ch. IX. Oxford: Clarendon Press. PAULY, H., PACKER, L. & SCHWAN, H. P. (1960). J. biophys. biochem. Cytol. I, 589. PAULY, H. & SCHWAN, H. P. (1959). 2. Naturforsch. 14b, 125. REDWOOD, W. R., TAKASHIMA, S., SCHWAN, H. P. &THOMPSON, T. E. (1972). Biochim. Biophys. Acta. 255, 557. SCHWAN, H. P. (1963). In Physical Techniques in Biological Research (W. L. Nastuk, ed.), Vol. 6. pp. 323407. New York: Academic Press Inc. WAGNER, K. W. (1914). Arch. Electrotech. 2, 371.

APPENDIX

A

Derivation of Equation (12) and Inequality

(13)

The complex fractional term of text equation (11) may be rewritten as

(AlI where z =jo 01= (anK.+IC,+l)/(anE.+E,+l)Ev B = @nK, + JL+ dl@nE,, + 8, + 1)~ Y = (anE,+E,+l)l(bn&,+E,+l).

642)

(A3)

Introducing Pn = Gsw,

and

P,,+~ = ~,+lI+.l~,

2 n+l

= &.+1 1%

Pn+l

= ~,+1/%,

(A4)

the constants, c1and fi, are further transformed: a = P,(u,+~L,+~)/(~,+~,+,)

On the other hand, combining

and

WI

the 4 relations defined in (A4), we have Pn+1 -=-. Pll

10

B = pn(b,+~“+l)/(b,+;t,+l).

&I+1 A II+1

W)

264

A.

Under the condition that

IRIMAJIRI

ET

AL.

(10) for i = n + 1, viz. pn+ i/p, 3 1, it follows from (A6)

Pn+1 2 -<-<-Sl,

a”+Pn+l

P”

> h+P”+l

&f&+1

(A71

b,+k+l

since b, > a, > 0 from text relation (8). Multiplying each term of (A7) by P,, followed by substitution therein of equations (A5), we.obtain inequality (13).

APPENDIX

B

Derivation of Equation (19) Considering the recurrent advantage of the mathematical written as x,-1

nature of text equation (3) we shall take inductive method. Equation (3) for i = n - 1 is

= c,-1 .a,*-,

.

4-l

+x,

bn-1+x,

which may be combined

with equations (9) and (15) to give x

_ n

= 1

c,n

.-z+Pn-l l

mz>

z+pnq.m

where

c;-1 = c,-Ia,-,>o,

Equation

mz> = P&) + aA44

032)

e,(z) = PA4 + BzQz(4

@3)

CI~= a,-,/ck>O

and

pz = b,-,/ck>O.

(B3) is further rewritten, using (B2), as &Cz> -= & + bQz(4

@4)

where 6, = pz-a,>o. Recalling definitions (15) and (16), therefore, we can predict the changes in the sign of Fz(z) and &(z), respectively, from (B2) and (B4). In Table 1 is summarized such a qualitative assessment. Here, -pk’s and - &‘s stand for the roots of equations, Fz(z) = 0 and p,(z) = 0, respectively. The result presented demonstrates that either of these equations has two negative roots (hence, j& >O, Lfk> 0) and that the possible relationship between the para-

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265

TABLE 1 Assessment for p,(z) and g2(z) in the range z < 0 (4 ql(=hl)
--z P2(4

41 Sl Pl

92

-

+

0

@2e,O

0

-

P*(z) 62&(z) Q2(z)

+ !

--z

P2(4

Pl

P2(4

1

Ql

41

P2

0

(1)

+

(2)

+

0

G2

P2

(3) =(l)+(2) (4) (5) = (3)+(4)

P2

0 cl +

0

0

0

41

P2

+

0 +

+

q2

q2

+0--o+

A

41

0 +

P2(z)

P2

0 1

q2

0

f

Pl

v2&)

0 0 -

-

0 +

s2~~~ 2=

--z

42

(P.-~ =ql orq,):

Pl

P2(4 ~2g2(4

0 0

(b) p1
+

s2%?22 p,(4 PA4

Pl

= (z+PJ(z+Pz)> = &4z+~1Xz+rJ*h

0 + +o-

-

0

0

-

0

-

Q&, = QA4

82

0 -o+

(1)

(2) (3)=U)+G9 (4) (5) = (3)+(4)

+

(1)

0 + 0

(2) y), =

(l)+(2)

(5) = (3)+(4)

(z+q&z+qJ =

Mz+%)(z+&h

meters, P,,- 1, &‘s and &‘s, all positive, ought to be found in the following sequences :

Now, define new polynomials (Z+Pn-l)‘F2(Z)=

of degree 3, P3(z) and &(z), such that C&

fi j=l

(Z+pj)-

&P,(Z)

036)

(B7)

266

A.

ET

IRIMAJIRI

AL

where a$ = 1 + tx2and F2 = 1 + B2. (Note that herepj refers to p,,- 1, p1 or Ji2 ; qj refers to pn-2, q1 or 42 ; and, in addition, the suffix j is so chosen as to indicate the increasing order Of pj’s or qj's.) Then all the sequences listed in (B5) can be lumped together into P1<41
As regards p,, - 2, a candidate for

qj’s,

h-2

z

Consequently,

@W

there are no restrictions other than Pn-1.

pne2 may take any place in sequence (BS), yielding

Again note that the pj’s and qj’s in sequences (B9) differ from pk's and qk's associated with the quadratic functions P2(z) and Q2(z) in text equation (15); we have just renamed the family of parameters implicit in equation (Bl) only for economy of symbols. At any rate, the sequences (B9) above obtained guarantee that x,,- r in general takes the form : (z +Pl)(z

&-'

+P2)cz

+P3)

P3(4

= "-1(z+ql)(z+q2)(z+q3)E

"-l

OW

Q3(z)

wherein the polynomials, P3(z) and Q3( z) , are relatively prime. If, however, p,, _ 2 coincides with p1 or p2, then those polynomials for x, _ i can not be of third degree but remain at the same degree as those for x,. The next step to come up will be the derivation of the general term Xi based on the assumption that the precedent term Xi+1 is formally given by

with a set of sequences, which is a formal extention of (B9): 4l
. .
P1<41442
.

.

.

.

Pl<41
.

.

.

.

.

. .
.

.

.

.

.

.

VW

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Now, equation (3) is combined with equations (9) and (Bll)

to give W3)

Here, analogously to definitions (B2) and (B4),

with cI,+l-i

= Ui/C;+l,

Pn+l-i

=

biic:+l

and dn+l-i

z Pn+l-i-C4n+l-i>O~

If the roots for the equations, p n+l-i(~) = Oand gn+rei(z) = 0, areallreal, negative and different, then we may write: n+l-i F n+l-iCZ> = “,-!-l-i n CZ+Pk), Pk+l>Pk>O k=l W5) n*l-i = B’,+l-i

&+l-i

n

tz+qk),

qk+l>qk>o

k=l

I

where a’,+i-i = l+tl,+i-i and 8:”+ i -i = 1 + j, + 1 _ i . Indeed, an elementary algebraic analysis, based on (B14) under the condition (B12), of changes in the signofFn+,+(z)andc “+ I -i(z) for z < 0, such as already presented in Table 1, reveals that all thePk’s and &‘s in expression (B15) are positive and in addition can be arranged in the following order of sequence: (j=k=l,2,...,n+l-i

pj
W6)

when one renames each member of the set of &‘s (inclusive of pi) as pi (j = 1,2, . . . , n + 2 - i) according to their magnitude. It is to be noted that sequence (B16), a generalized form of inequalities (13) and (B8), plays a fundamental role in performing the present analysis. Thus it is feasible, by renaming the &‘s and pi- 1 collectively as q’s appropriately suffixed, to extend (B12) to the following set of sequences : For Pi-l(=ql)
qj
Pl<4J<4l+l
(j= 1,2,...,1-1) For Pn+2-i
(j=l,2,...,fZ+l-i) (z=l,&...,n+l-i): Pj<4j+lcPj+1

(j=1+1,1+2,...,n+l-i) Pj
@17) I

A.

268

IRIMAJIRI

ET

AL.

Under the condition of (B17), relations (B15) are allowed to substitute for the right-hand side of equation (B13), yielding the general expression: , (z+P1)(z+PZ). xi

=

Ci(Z+ql)(Zfq,).

. .Cz+PnA2-i)

_

. s(Z+q,+z-i)

=

, pn+2-i(z) ci

Q,+z-i(Z)

0318)

where C; = ci&&+i-i/A+1-i. Now, changing the suffix i to 1 in equation (B18), we can immediately obtain text equation (19). It should be noticed here again that, if pi-1 coincides with any one of&‘s of (B15), then the degree of the polynomials for Xi will be degraded to n + 1 - i. This is the reason for the statement given under item (c) of section 2(C). APPENDIX

C

Proof for Equation (23) and Sequence (24) Owing to the similarity between text expressions (3) and (20), much the same logic as employed in deriving equation (B18) applies to the derivation of equation (23) and related sequence (24). More specifically, it is sufficient for the present purpose to show a possible correspondence between the symbols of equation (3) and those of equation (20). Putting i = 0 in equation (3), we formally obtain x0

=

c,L$

ao+x1 ___ bo+x,

or, when rewriting x0 and A,* in keeping with definitions E;f; -=~o---..-- E;S a,+x, ET1 E?!, b,,+x,

(Cl) (5) and (6), (C2)

which simply reduces to ET, =

coeg

ao+x1

cc31

bo+xl

since E? r is trivial and highly arbitrary, and hence it may be taken to be unity, for instance. It then follows that E-~ = 1 and Now comparison

K-~ =O.

W4)

of equations (C3) and (20) leads to the statement that

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both equations should become homologous, given the following correspondence : 3*. ‘~,=a, Eo-E) b,=b, and co=c, or $o=@.

set of

(C5) Consequently, if we define an additional symbol such that p _ r = 0 which is consistent with (C4), text equation (22) becomes identical with equation (B13) for i = 0. Then it is straightforward from the result of Appendix B to write down text equation (23) in place of equation (B18) with i = 0. Sequence (24) is readily obtainable from (B17) because, in the present case, invariably pi-1 = 41 = 0.

APPENDIX

W

D

Proof for Inequality

(26)

Equating the right-hand sides of equations (23) and (25), and rearranging, n+2

n+2 C” k;l

(Z+Pk)

=

n+2

C” kj-$z+qk)+k~l

n+2 (Z+qk)

k;l

&

“ff@+qk)

CD11

k=l kZJ

where c” = ~‘(1 +a’)/(1 +b*). For z = -ql, n+2 cu kgl

k#J

therefore, we have n+2

(Pk-qJ)

=

ri

n

W’)

(qk-qJ),

k=l k#J

whence n+2 ‘J

=

,

c”(PJ-qJ)k~l

I=

1,2 ,...,

?2+2.

(D3)

k+J

Text sequence (24) may be rewritten as and

CD41

so that the terms, (pk - qr) and (qk in equation (D3) are of the same sign for every combination of k and 1.Thus it is clear from (D3) and (D4) that rk > 0 foreveryk= 1,2,..., n + 2, since C”> 0 by definition. qJ),